Finite Group Databasebeta

← #10591310

Subgroups — #10591310

Order 576.

470 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Dic6C211yesmaximal, characteristic, complemented
2C2×#3752C211yesmaximal, characteristic
3C2×#3722C212yesmaximal
4C3×C3⋊#95C211yesmaximal, characteristic, complemented
5#63579C212yesmaximal, complemented
6#5231—31nomaximal
7#4673—31nomaximal
8C23×C3⋊D3C2211yescharacteristic
9C22×C32×C4C2211yescharacteristic
10C22×C3⋊Dic3C2211yescharacteristic
11C22×C3×Dic3C2212yes—
12C22×C3×Dic3C2212yes—
13C2×C3×Dic6—22no—
14C2×C3×Dic6—24no—
15#3752—22no—
16#3722—24no—
17C22×C3×Q8—31no—
18C22×Dic6D311yescharacteristic, complemented
19C2×(C2×D3)⋊C4—31no—
20C2×(C2×D3)⋊C4—31no—
21C2×(C2×D3)⋊C4—32no—
22C2×(C2×D3)⋊C4—32no—
23C2×(C2×D3)⋊C4—61no—
24C3×#95—31no—
25C3⋊#95—32no—
26C3⋊#95—32no—
27C3⋊#95D311yescharacteristic, complemented
28C23×C32C2311yescharacteristic
29C22×C3⋊D3—21no—
30C22×C3⋊D3—21no—
31C22×C3⋊D3—22no—
32C22×C3⋊D3—22no—
33C22×C3⋊D3Q812yescomplemented
34C2×C32×C4—22no—
35C2×C32×C4D412yes—
36C2×C3⋊Dic3—21no—
37C2×C3⋊Dic3—22no—
38C2×C3×Dic3—22no—
39C2×C3×Dic3—22no—
40C2×C3×Dic3—22no—
41C2×C3×Dic3—22no—
42C2×C3×Dic3—22no—
43C2×C3×Dic3D414yes—
44C3×Dic6—28no—
45#393—91noSylow
46C23×D3—31no—
47C23×D3—31no—
48C23×D3—61no—
49C2×C3×Q8—64no—
50C2×C3×Q8—62no—
51C22×C3×C4—21no—
52C22×C3×C4—32no—
53C22×C3×C4—32no—
54C22×C3×C4C2×D311yescharacteristic
55C22×C3×C4C2×D311yescharacteristic
56C22×Dic3—31no—
57C22×Dic3—31no—
58C22×Dic3—61no—
59C22×Dic3C2×D312yes—
60C22×Dic3C2×D312yes—
61C2×Dic6—22no—
62C2×Dic6—24no—
63(C2×D3)⋊C4—64no—
64(C2×D3)⋊C4—64no—
65(C2×D3)⋊C4—62no—
66(C2×D3)⋊C4—62no—
67(C2×D3)⋊C4—64no—
68C22×C32C2×D411yescharacteristic
69C22×C32C2×D412yes—
70C22×C32C2×Q811yescharacteristic
71C22×C32#4112yes—
72C22×C32#4111yescharacteristic
73C2×C3⋊D3—22no—
74C2×C3⋊D3—24no—
75C2×C3⋊D3—41no—
76C2×C3⋊D3—41no—
77C2×C3⋊D3—42no—
78C32×C4—22no—
79C3⋊Dic3—41no—
80C3×Dic3—24no—
81C3×Dic3—42no—
82C22×Q8—31no—
83C2×C22⋊C4—92no—
84C2×C22⋊C4—91no—
85#95—31no—
86#95—92no—
87C23×C3—21no—
88C23×C3C22×D311yescharacteristic
89C23×C3C22×D311yescharacteristic
90C22×D3—32no—
91C22×D3—32no—
92C22×D3—62no—
93C22×D3—62no—
94C22×D3—62no—
95C22×D3—61no—
96C22×D3—62no—
97C22×D3—62no—
98C22×D3—61no—
99C22×D3—61no—
100C22×D3—62no—
101C22×D3—61no—
102C22×D3—62no—
103C22×D3—122no—
104C22×D3—122no—
105C3×Q8—68no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—34no—
110C2×C3×C4—42no—
111C2×C3×C4—62no—
112C2×C3×C4—62no—
113C2×C3×C4—62no—
114C2×C3×C4—62no—
115C2×C3×C4—62no—
116C2×C3×C4D1212yes—
117C2×C3×C4#6712yes—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—62no—
124C2×Dic3—61no—
125C2×Dic3—62no—
126C2×Dic3—62no—
127C2×Dic3—61no—
128C2×Dic3—122no—
129C2×Dic3#6714yes—
130Dic6—28no—
131C2×C32#12011yescharacteristic
132C2×C32C22⋊Q812yes—
133C2×C32C22⋊Q811yescharacteristic
134C2×C32#12411yescharacteristic
135C2×C32#12412yes—
136C3⋊D3—42no—
137C24—91no—
138C2×Q8—64no—
139C2×Q8—62no—
140C22×C4—32no—
141C22×C4—32no—
142C22×C4—91no—
143C22×C4D3211yescharacteristic
144C22⋊C4—184no—
145C22⋊C4—182no—
146C22×C3—21no—
147C22×C3—21no—
148C22×C3—22no—
149C22×C3—21no—
150C22×C3—22no—
151C22×C3C2×D1211yescharacteristic
152C22×C3#23512yes—
153C22×C3D3×D412yes—
154C22×C3#23711yescharacteristic
155C22×C3D3×Q811yescharacteristic
156C22×C3D3×Q811yescharacteristic
157C22×C3#23911yescharacteristic
158C22×C3#23912yes—
159C22×C3C2×#6711yescharacteristic
160C22×C3C2×#6712yes—
161C2×D3—64no—
162C2×D3—64no—
163C2×D3—62no—
164C2×D3—62no—
165C2×D3—64no—
166C2×D3—124no—
167C2×D3—122no—
168C2×D3—124no—
169C2×D3—122no—
170C2×D3—121no—
171C2×D3—121no—
172C2×D3—122no—
173C2×D3—121no—
174C2×D3—122no—
175C2×D3—121no—
176C3×C4—22no—
177C3×C4—22no—
178C3×C4—42no—
179C3×C4—64no—
180C3×C4—122no—
181Dic3—24no—
182Dic3—42no—
183Dic3—122no—
184Dic3—121no—
185Dic3—121no—
186C32#39311yescharacteristic, complemented, Sylow
187C23—92no—
188C23—181no—
189C23—182no—
190C23—181no—
191C23—182no—
192C23C2×D3211yescharacteristic
193Q8—68no—
194C2×C4—22no—
195C2×C4—34no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—182no—
202C2×C4—181no—
203C2×C4#62612yes—
204C2×C3—21no—
205C2×C3—21no—
206C2×C3—22no—
207C2×C3—22no—
208C2×C3—21no—
209C2×C3#87511yescharacteristic
210C2×C3#88211yescharacteristic
211C2×C3#88512yes—
212C2×C3#89612yes—
213C2×C3#89711yescharacteristic
214C2×C3#94611yescharacteristic
215C2×C3#94612yes—
216C2×C3#94711yescharacteristic
217C2×C3#94712yes—
218C2×C3#95311yescharacteristic
219D3—124no—
220D3—122no—
221D3—122no—
222C22—182no—
223C22—184no—
224C22—361no—
225C22—362no—
226C22—361no—
227C22#379611yescharacteristic
228C22#379711yescharacteristic
229C22#379912yes—
230C22C2×#62611yescharacteristic
231C22D3×#6712yes—
232C4—22no—
233C4—64no—
234C4—122no—
235C4—361no—
236C3—21no—
237C3#467311yescharacteristic, complemented
238C3#523111yescharacteristic, complemented
239C2—362no—
240C2#6386411yescharacteristic
241C2#6387111yescharacteristic
242C2#6388812yes—
243C2#6390712yes—
244C2#6397711yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
FittingC22×C32×C4C22
prime radicalC23×C3⋊D3C22
hypercenterC22×C4D32
hypocenterC32#393
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×C3×Dic6C2semidirect
4. C3×C3⋊#95C2semidirect
5. #63579C2semidirect
18. C22×Dic6D3semidirect
27. C3⋊#95D3semidirect
33. C22×C3⋊D3Q8semidirect
186. C32#393semidirect
237. C3#4673semidirect
238. C3#5231semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3×Dic62
C2×#37522
C2×#37222
C3×C3⋊#952
#635792
#52313
#46733